Phyllotaxis
the spirals are counted, not admired
Concept

Hop length — where it appears

How far it is across the surface from an organ to the one a stated number of places above it. Ordering the lags by it gives a lattice's contact families at the top, and the family a cut leaves standing is sometimes the shortest and sometimes the second.

Named by 15 essays across 3 fields — each of them below, with the objects they name alongside it.

Hops give 55, 34, 89, 21 and walls give 34, 55, 21, 89. The leading families of a 900-point head, ranked twice from the same points. On the left, by the median hop the lag makes divided by the local spacing — the measurement a parastichy count is; on the right, by the share of the 1,732 interior walls the lag carries. The lists hold the same four numbers and four of them change place. The lines between are the permutation, and it is why a family read off a wall count is not a family read off a hop length.

Two rankings, one list

An essay in this collection claimed that the four shortest index hops on a seed head and the four largest shares of its cell walls are the same four numbers in the same order, and called the correspondence exact. Measured again from the same points, the two lists hold the same four families and order them differently, and they order them differently in five of the six bands the head can be read in.

tissue · Neighbour definition How often a cell on a 2,400-organ golden head has a wall the contact cut gets wrong, by its distance from the nearest flip ring. The 1,680 interior cells of a 2,400-organ golden head, grouped by distance from the nearest flip ring in twentieths of the head's wall spacing, 1.92. The bars are the share of each group with at least one wall a three-family contact cut gets wrong; the dots are the share with five or seven sides. 353 cells have such a wall and the farthest is 0.638 spacings from a ring. Of the 1,292 cells 0.66 spacings or more from every ring, none has.

The empty interval is the rings

No single cut-off on hop ratio turns the contacts a count keeps into the walls a tessellation draws, on any whole head at any size. Read cell by cell against the flip rings the divergence angle puts in closed form, every disputed cell lies within two thirds of a wall spacing of a ring, and with one spacing either side set aside a single cut-off between 1.430 and 1.444 serves every golden head from 900 organs to 9,000.

tissue · Neighbour definition The centre of a 900-organ Lucas head, with the band of two thirds of a spacing drawn round every flip ring. The organs of a 900-organ Lucas head out to a radius of 12.5, with a pale annulus 0.66 of a wall spacing either side of each flip ring, the rings keeping 1, 3, 4, 7, 11, 18. Warm: cells with five or seven sides; dark: six-sided cells a three-family contact cut gets wrong. Ringed: the one cell well inside a band that the cut reads exactly, organ 17 at radius 4.12, 0.23 of a spacing from its ring. Where two annuli overlap, a cell can sit well away from its nearest ring and still inside the next one's band.

The blur was at the centre

On a Lucas head the band of disputed cells round each flip ring looked blurred at its inner edge — exact cells as close as 0.23 of a wall spacing, disputed hexagons out to 0.59 where a golden head's stop at 0.43. Read a ring at a time, the two heads carry the same band on every resolved ring, to a hundredth: disputed hexagons within 0.16, exact cells from 0.64, a ring's own number of fives and of sevens and the number before it of hexagons. Every difference is inside a radius of six, where the Lucas rings of 4, 7 and 11 sit closer together than the band is wide, and the one exact cell is organ 17, which has no organ eighteen behind it.

tissue · Neighbour definition A 900-organ golden head displaced by 0.02 of a wall spacing, with its flip rings and the cells in dispute. Every organ of a 900-organ golden head inside the rim cut, each moved by a seeded gaussian displacement of 0.02 of a wall spacing, with the flip rings the divergence angle puts at radii 11.3, 18.2. Warm: the 150 cells with five or seven sides, 13 of them more than 0.66 of a spacing from every ring. Dark: the 59 six-sided cells with a wall a three-family contact cut gets wrong. Of the cells a spacing or more from every ring, 0 are disputed.

A hundredth of a spacing

Off the flip rings one hop-ratio cut-off turns a seed head's counted contacts into its cell walls, on every head from 900 organs to 9,000. Displace the organs and it is the first thing to go: shut by a fiftieth of a wall spacing on 900 organs and a two-hundredth on 9,000, because it is decided by the worst of thousands of cells. The three-family count survives two to four times further, because each cell only has to beat its own margin, and the rings keep their fives and sevens in between. All three fail from the rim inward, since the margin one spacing from a ring is 9.7 divided by the ring's family number.

tissue · Neighbour definition The disputed cells round the ring of 55 on a 2,400-organ head twisted by 0.2 radians at the rim. Every organ turned about the centre by 0.2 times its radius over the head's. Bars: the six-sided cells a three-family contact cut disputes (grey-blue) and the five- and seven-sided cells (warm) within two spacings of the ring the untwisted divergence puts at 55, by signed distance from it. There are 34 disputed hexagons, with a median at 0.54 of a spacing, and 110 fives and sevens. The dashed line is where the twisted divergence puts the ring in closed form, 0.54 of a spacing out; the band keeps its cells and its width and sits on it.

The band moves, it does not blur

Displaced organ by organ, a seed head loses its single contact cut-off first, its rings' hold on their fives and sevens next and its three-family count last. Displaced by a smooth field that moves neighbours together, the same head keeps its census — the same 353 disputed cells and 264 fives and sevens at every step up to a third of a spacing — and moves the band instead. A twist moves each flip ring exactly to where the twisted divergence puts its tie, the ring of 55 by 0.53 of a spacing, the ring of 34 the other way. Read against strain, correlation helps the cut-off and not the count, and on a 900-organ head the two fail at the same step: the order was an order of blurring.

tissue · Neighbour definition The hops the correction is fitted over, with the new one at the near end. Each cluster of rows placed by the lag it kept and the angle of the hop that lag keeps. The four the correction was fitted over run from 19.5 to 39.1 degrees; the new one sits at 12.78 degrees, a third smaller than any of them. A fifth point beyond the near end of a fitted range is a test of the fit, where a fifth point between two old ones would mostly have been a restatement.

A fifth cluster

A correction to the exchange's size was fitted over four hop clusters and its own file said so. A search turned up a fifth, at a hop smaller than any of the four, and the rule is right on it — which is what a prediction being confirmed looks like when the confirmation is worth having.

cylinder · Ablation g005 at offset 6, read against two levels. One wrecked cut's four residue classes, drawn twice. On the left the level is the median of the four class means, which falls on the class at -5.5 degrees sitting on its own; three classes are then exceptions and the row is set aside as having no balanced pair, summing to -173.6 degrees. On the right the level is the pair of classes that agree, at -138.6 degrees; two classes are then exceptions, they are adjacent and equal and opposite at 133.1 and 134.2 degrees, and the row is an exchange. The exchange table goes from 20 rows to 21.

The twenty-first row

Recomputing the level moves one row out of the set the exchange sets aside and into the exchange itself. Its hop is four times larger than any the correction was fitted over, and the correction fails on it in the one way it had never failed.

mechanism · Damage shape Twenty rises at the fine end of both branches, cut at every offset. One row per rise searched, coarse at the top of each block. The bar names the counted pair the stem shows and the numbers on the right are the lags its wrecking cuts leave standing. A pale row is a rise whose settled divergence has left the branch it was started from by more than 20 degrees, which is what happens below the ladder's finest rung — the pairs there are 2 and 4, 8 and 16, 11 and 22, which are not two consecutive terms of any additive sequence. One rise on the Lucas branch keeps a lag of 11 while still on it.

One rise below the census

Ten lattices were cut at every offset and their surviving lags came back as four numbers. One rise further down a rung the census already sweeps, three cuts keep a lag of eleven — which is a fifth number, on a lattice nothing about was unusual except that nobody had cut it.

cylinder · Ablation Twenty rises at the fine end of both branches, cut at every offset. One row per rise searched, coarse at the top of each block. The bar names the counted pair the stem shows and the numbers on the right are the lags its wrecking cuts leave standing. A pale row is a rise whose settled divergence has left the branch it was started from by more than 20 degrees, which is what happens below the ladder's finest rung — the pairs there are 2 and 4, 8 and 16, 11 and 22, which are not two consecutive terms of any additive sequence. One rise on the Lucas branch keeps a lag of 11 while still on it.

The lag that never survives

A correction to the exchange's size rests on four hop clusters, and a fifth would be the first real test of it. The prediction was written for a golden lattice at a lag of eleven. No golden lattice on this ladder reaches one, and the reason is a fact about the rule rather than about the search.

cylinder · Ablation The 17 rows of the exchange table, gathered by the lag they kept. One bar per surviving lag, its length the number of rows the table holds at that lag, with the hop that lag's stems keep written beside it. The hop is nearly constant inside a lag, so a correction fitted over 17 rows is fitted over four hops — which is the denominator that matters and is much smaller than the row count suggests. Adding a lag to the table is worth more than adding rows at a lag already in it.

A family that is a multiple

When a wrecked cut keeps a family that is not one of the lattice's counted pair, the first case looked like a rule: it was half of one of them. The second case is four times the other, which makes the rule a coincidence and leaves a weaker statement that is probably the true one.

cylinder · Rung interior The Lucas 7/11 rung between 0.007 and 0.006, cut at every rise. One row per offset, one column per rise of the search, coarse on the left. A pale cell is a rise at which that offset's cut recovers; a dark cell is a cut that wrecks and keeps a lag of 7; a warm cell is one that keeps 11. The nine intermediate rises of the ten-thousandth grid were cut first and the nine of the hundred-thousandth grid inside the one bracket they opened. The change sits between 0.0064 and 0.00639, one step of the grid the ladder names its rises on, and above it no cut keeps 11 anywhere.

One step of the grid, again

A search of the fine end stepped from one rise to another a hundred grid steps away, and the family a cut keeps changed somewhere between. Cutting the rises in between puts the change inside one step, with the lattice identical on both sides.

cylinder · Ablation The two shortest hops across the Lucas 7/11 rung, and where they cross. The lengths of the two shortest hops at every one of the 378 rises of the rung, fine on the left, computed from the settled divergence rather than measured off a cut. They cross exactly once, between 0.00804 and 0.00803, which is this rung's handover. The rise at which a cut first keeps the longer family is 164 steps of the grid further down and is marked separately; nothing happens to either length there.

The hops cross once

Walking a whole rung at the grid its rises are named on costs a few hundred stems and no cuts at all, and it answers a question nobody had asked: whether a rung has one handover or several. It has one, and the ladder had recorded it in the wrong place.

cylinder · Ablation Two rises on the Lucas 7/11 rung, each located to one step of the grid. The whole of the Lucas 7/11 rung, coarse on the left, with the two rises this round located. The two contact steps change places between 0.00804 and 0.00803, found by walking all 378 rises of the rung and evaluating hop lengths, with no cut stems at all. The family a cut leaves standing changes between 0.0064 and 0.00639, found by cutting 89 stems. They are 164 steps of the grid apart, which is 45 per cent of the rung's whole span, so the geometry crossing is not what moves the survivor.

Two rises far apart

The whole handover thread rests on the claim that where the two contact steps change places is not where the survivor does. On one rung both rises are now located to a single step of the grid, and they sit forty-five per cent of the rung apart.

cylinder · Ablation Six rungs walked at the grid, 10 crossings between them. Each row is one rung, drawn from its coarse end on the left to its fine end on the right and scaled to its own width so positions inside different rungs can be compared. The shaded stretch is the band the ladder grows around the rise it recorded as that rung's handover. five of the six rungs carry exactly one crossing, and on each of those the whole stretch a second one could sit in has been walked at the grid and closed at both ends. The 3/4 rung of the Lucas branch carries five, spaced 22, 12, 15 and 20 grid steps apart, and its band holds three of them. The rule on each row is a located crossing.

Five rungs walked

Six rungs of the ladder carry a handover and only one of them had ever been walked at the resolution its rises are named on. Walking the other five costs 1,224 grown stems and no cuts at all, and it returns a crossing count per rung — five ones and a five.

cylinder · Handover grid A reading that steps against an ordering that slides, on six rungs. Every hop length here is arithmetic on the settled divergence, and the settled divergence is a mean of angles read off a lattice placed on a fixed set of azimuths — so it is read in steps rather than continuously. Between two steps the ordering slides with the rise; at a step it jumps. The bar is how much a jump is worth in slides on each rung, measured inside the stretch that was walked at the grid so that the six are read over comparable regions. The account is a threshold at one and it is right on all six: the 3/4 rung sits at 12.71 and carries five crossings, and on three rungs the reading does not step inside the window at all.

One crossing or two

One rung of the ladder changes hands five times where the other five change hands once, and the difference is not in the geometry. It is a divergence read in steps against an ordering that slides, and the account is a threshold at one that is right on all six rungs.

cylinder · Handover grid

Named alongside it

The objects these essays reach for when they reach for this one.

Contact familyHonest limitsClaim testingResolutionAblationNegative resultRungInstrument settingCut-offDefect ringDiscretisationHandover

All concepts